# GLIP Responsivity

Equation (50.1) for the GLIPs with not too large (realistic) number of the GLs in the GLIP part *(N* *R _{(}„<^ j*

_{9}*h*

_{0t0}/ hool.

with

Here, the factor £ is determined by conditions of reflection of the incident radiation from the GLIP top interface [14]. Equation (50.6) turns to that derived and used previously for GLIPs with the undoped barriers and inner GLs [20, 21] at e_{GL} = 0 and Z_{B} = 0 when £_{GL} is replaced by *V/{* yjp + *N)d.* For example, setting | = 1 *,p =* 0.01, k_{b} =

5, and Д = 0.1 - 0.5 eV, for the characteristic responsivity we obtain Д ^ (2 - 10) A/W.

The value of the GLIP responsivity given by Eqs. (50.5) and (50.6) corresponds to the photoconductive gain*g = [p[ j** ^{/2}* + Л/)]'

^{1}—

*[pN*

*]'*(compared with Refs. [24, 25, 29, 30]). The origin of this gain is associated with the accumulation ofthe charges formed in the GLs by the photogenerated holes. The latter is due to a much more effective confinement of the photogenerated holes than the photogenerated (photoexcited) electrons that stems from the condition Д < A

^{1 }_{v }accepted in Section 50.2. If this condition is violated, the escape probability of the photogenerated holes from the GLs can become rather high leading to vanishing of the photoconducting gain effect.

**Figure 50.3 **Spectral characteristics of GLIPs with Д **= **0.1 meV, five undoped GLs (X_{GL} = 0), and different donor and acceptor densities in the barriers X_{B} (in units 10^{12}crrr^{2}) at *T =* 100 K. The dashed line corresponds to GLIP with doped GLs (e_{GL} = 0.1 eV, X_{GL} = 0.8 x 10^{12} cm^{-1}) - curves for different barrier doping are undistinguished.

Figures 50.3 and 50.4 show the responsivity calculated using Eq. (50.6) with Eqs. (50.4) and (50.5) as a function of the photon energy for the GLIPs with different barrier heights Д and different doping levels of the inter-GL barriers and GLs. For the definiteness, the following general parameters are assumed: *N* = 5, £ = 1, *p* = 0.01, *d _{Gl}* = 2 nm, r

_{esc}/r

_{relax}= 0.1,

*T*= 100 K, and

*U = V/ dE*'

_{tunn}= 0.5.

*m*

*=*(0.14 - 0.28)m

_{0}(m

_{0}is the mass of bare electrons) and

*d*= 10 nm. The value

*U*= 0.5 corresponds to

*V/N*= 0.026 - 0.038 V at Д = 0.1 eV and

*V/N*= 0.11 - 0.15 V at Д = 0.25 eV.

One can see that an increase in the barrier doping level, which results in higher tunneling transparency of the barrier for the photoexcited electrons and, hence, higher probability of their escape, leads to a substantial increase in the responsivity at relatively low photon energies *(hco* < 2Д). The responsivity of the GLIPs with smaller Д is higher than that of the GLIPs with larger Д in the low photon energy range (compare the curves in Figs. 50.3 and 50.4). An increase in the responsivity at relatively low photon energies exhibited by the curve for X_{B} = 0.3 x 10^{12} cm"^{2} in Fig. 50.4 is attributed to the factor *1/hcoin* Eq. (50.6) (see also a comment in Section 50.6). Marked values of the responsivity in the range *hco *S 0.05 eV (about several A/W, as seen from Figs. 50.3 and 50.4) imply that the GLIPs with properly doped barrier layers can operate not only in near- and mid-infrared spectral ranges but also in the terahertz range.

**Figure 50.4 **The same as in Fig. 50.3 but for GLIPs with Д = 0.25 eV.

The GL doping by acceptors also modifies the responsivity spectral dependence: its increase (and, therefore, increase in the Fermi energy e_{GL}) gives rise to a marked shift of this dependence toward higher photon energies (compare the solid and dashed lines in Figs. 50.3 and 50.4). In the case of doped GLs, the barrier doping weakly affects the spectral dependence in question (the curves corresponding to different values of X_{B} and f_{GL} = 0.1 eV are practically merged).

In principle, the temperature smearing of the electron energy distributions in the GLs somewhat affects the photon absorption probability at *hco* ^ 2(Д + £_{Gl}) due to the degeneracy of the electron system near the Fermi level. However, the variation of the temperature in the range *T* = 50-200 only slightly changes the above spectral dependences.

*Effect of Doping on the Characteristics of Infrared Photodetectors*

920

# GUP Detectivity

The dark current limited detectivity is usually determined as

where *g* is the photoconductive gain which was introduced in Section 50.4. The dark current in the GLIPs is determined by the tunneling of the thermalized electrons from the GLs (amplified by the electron injection from the emitter GL). One can assume that the main contribution to this tunneling is provided by the electrons with the energies close to the Fermi level. Due to the specific features of the tunneling barrier shape, the tunneling exponent depends on the barrier parameters Д_{в} and *d.* Considering this and generalizing the pertinent equations obtained for the GLIPs with the undoped barriers and inner GLs, one can use the following relationship for the dark current:

where for Д_{СЕ} = Д

*jmax* >^{s} the maximum current density which can be extracted from the emitter GL, and /_{E} is the pre-exponential factor, which depends on the emitter ideality factor *y _{E}.* Disregarding for brevity the effects associated with the emitter nonideality (analyzed previously [20, 31]), we put in the following = 0 and/

_{E}= 1.

At elevated temperatures, the thermionic escape of electrons from the GLs can also contribute to the dark current. The pertinent dark current density can be presented as

where the quantity Д,_{Ьетт} Plays the role of the thermionic activation energy for the thermalized electrons in the GLs and *c* ~1. Generally, A “ AgL ^{+ e}GL ~ Atherm S A + £_{G}L-

Considering for simplicity the interpolation formula for the net dark current density in which the contributions given by Eqs. (50.9) and (50.12) are summarized, we arrive at the following equation for the dark current limited GLIP detectivity:

with

Here

is the quantity characterizing the relative contribution of the tunneling and thermionic processes. For the GLIPs with undoped barrier setting *E _{GL} = V/dN, F* = 0, and Atherm — A + e

_{GL}, Eq. (50.15) yields

Assuming *N* = 5 and ;_{max} = 1.6 x (10^{s} - 10^{6}) А/cm^{2}, at the same other parameters as in the above estimate of the characteristic responsivity *R,* we obtain *D* ^* (0.5 - 7) x 10^{s} cm VHz/W. Due to a large first exponential factor in Eq. (50.13), the real detectivity А» » *D*.* The GLIPs with a larger number of the inner GLs *N* can exhibit higher values of the dark current limited detectivity [because *D** °c [n (Ref. [33])]. The values of ;_{max} used here correspond, in particular, to the electron density in the emitter GL Z_{E} = 10^{12}cnr^{2 }and the try-to-escapetime r_{esc} = 10"^{13}- 10"^{12}s.

According to Eq. (50.13), the spectral dependence of the detectivity repeats that of the responsivity (shown, in particular, in. Figs. 50.3 and 50.4). The dipole doping of the barrier layers leads to an increase in the GLIP responsivity (primarily in the range of relatively low photon energies) but simultaneously to an increase of the dark current and, hence, a drop of the detectivity. The doping of GLs by acceptors, which modifies the spectral characteristics, promotes the dark current lowering and a rise of the detectivity. In principle, carefully choosing the levels of both types of doping, one can expect the optimal relationship between the responsivity and the detectivity. However, taking into account the fact that realization of both types of doping in one device can markedly complicate its fabrication, we restrict ourselves by considering the detectivity of the GLIPs with the doping of the GLs only. Therefore, we focus on the GLIP detectivity as a function of the GL doping and the temperature assuming that the barrier layers are undoped.

Figure 50.5 shows the detectivity of the GLIPs with undoped barrier layers calculated using Eqs. (50.13) and (50.16) as a function the Fermi energy £_{GL} (which is determined by the acceptor density in the GLs) for different temperatures. Figures 50.6 and 50.7 demonstrate how the detectivity of the GLIPs with different barrier heights Д and Fermi energy f_{GL} (i.e., different acceptor densities in the GLs S_{GL}) at different photon energy *hco* varies with increasing temperature *T.* We set *N =* 5, t_{esc}/r_{relax} = 0.1, and *U* = 0.5. The Fermi energy £_{gl} changes from zero to 0.1 eV in the acceptor density range S_{G}i = (0 — 8) x 10^{32} cm^{-2}.

As seen from Fig. 50.5, the detectivity *D**[ _{0}* is a nonmonotonic function of the Fermi energy £

_{GL}and the acceptor density Ic

_{L}in the GLs with pronounced maxima at certain values fc

_{L}and S

_{GL}. A pronounced increase in the detectivity is attributed to an increase in the barrier height for the thermalized electrons A + f

_{GL}with increasing doping level (see Fig. 50.2) leading to a diminishing of the tunneling and thermionic electron escape and, consequently, to a weaker dark current.

Figure 50.5 GLIP detectivity for Д = 0.1 eV and undoped barriers (Z_{B} = 0) as a function of the Fermi energy e_{0L} (acceptor density in GLs X_{GL}) for different photon energies *h*ft)at (a) *T* = 20 K, (b) *T* = 50 K, and (с) *T =* 80 К : 1—*h(0 =* 0.05 eV, 2-0.1 eV, 3-/ift)= 0.15 eV, and 4-0.25 eV.

A steep detectivity roll-off at increased acceptor densities is associated with the Pauli principle leading to an abrupt drop of the photon absorption and, hence, the responsivity when £gl becomes close to or larger than *hco/2.* Some difference in the steepness of the detectivity roll-off seen in Figs. 50.5a-c is due to a stronger smearing of the Fermi-Dirac distribution in the GLs at higher temperatures. The dependences shown in Figs. 50.5-50.7 indicate a marked decrease in the detectivity maximum with increasing temperature. This is explained by an increase in the role of thermionic processes at elevated temperatures.

Figure 50.6 Temperature dependences of detectivity of GLIPs with Д= 0.1 eV, undoped barriers and different Fermi energies c_{GL} for (a) *h(0=* 0.1 eV and (b) *h(0= 0.2* eV.

924

*Effect of Doping on the Characteristics of Infrared Photodetectors*

Figure 50.7 The same as in Fig. 50.6 but for Д = 0.25 eV and (a) *h(0 =* 0.25 eV and (b) *h(0=* 0.5 eV.

One needs to point out that the values of the detectivity *D**' _{(0 }*at certain values of the Fermi energies (acceptor densities) can be rather high. Taking into account the values of

*D**obtained in the above estimate, for the

*D'*maximum we find max

_{fi)}*D'*> 10

_{fi)}^{9}cm VHz/W.

One can see from Figs. 50.6 and 50.7 that the detectivity being a flat function of the temperature steeply drops at T exceeding a certain temperature:

This is associated with the inclusion of the thermionic contribution to the electron escape from the GLs. Although the enhancement of the GL doping results in a pronounced increase in the detectivity, it leads to a shrinking of the temperature range where *D** _{(0}* [and crossings of the curves in Figs. 50.6 and 50.7]. Indeed, Eq. (50.17) yields

*T*~ Д

_{therm}^{3/2}/(Д + f

_{GL})

^{1/2}, ie., a decreasing

*T*versus f

_{therm}_{GL }relationship.